Math 2.0

Probability theory and stochastic processes Definition

Semimartingale

Builds on FiltrationAdapted processLocal martingale

Definition

Let (Ω,ℱ,P)(\Omega, \mathcal{F}, P) be a probability space with a filtration (ℱt)t≥0(\mathcal{F}_t)_{t \ge 0} that is right-continuous and complete. A function a ⁣:[0,∞)→ℝa \colon [0, \infty) \to \R has finite variation if sup⁡∑k∣a(tk)−a(tk−1)∣<∞\sup \sum_{k} \abs{a(t_k) - a(t_{k-1})} < \infty on each interval [0,t][0, t], the supremum being over finite partitions 0=t0<⋯<tm=t0 = t_0 < \dots < t_m = t. A real adapted process (Xt)t≥0(X_t)_{t \ge 0} with càdlàg paths is a semimartingale if it can be written as Xt=X0+Mt+At,t≥0,X_t = X_0 + M_t + A_t, \qquad t \ge 0, where:

  1. (Mt)(M_t) is a local martingale with M0=0M_0 = 0.
  2. (At)(A_t) is an adapted càdlàg process with A0=0A_0 = 0 whose paths have finite variation almost surely.

Remarks

Brownian motion with drift, Xt=Wt+μtX_t = W_t + \mu t with WW a Wiener process and μ∈ℝ\mu \in \R, is a semimartingale: M=WM = W is a martingale and At=μtA_t = \mu t has finite variation. Fractional Brownian motion with Hurst index H≠1/2H \ne 1/2 is not a semimartingale for its own filtration, so it admits no such decomposition. The decomposition is not unique in general; it is unique when AA is required to be predictable and such a decomposition exists, as for continuous semimartingales. By the Bichteler–Dellacherie theorem, semimartingales are exactly the good integrators: the adapted càdlàg processes for which the stochastic integral of simple predictable processes extends continuously. Every semimartingale has a quadratic variation, and Itô's lemma holds for C2C^2 functions of semimartingales.

Reading path

To understand this page from the ground up, read these 53 pages in order. The foundations come first.

  1. Adapted process concept
  2. Construction of the reals from Cauchy sequences concept
  3. Element definition
  4. Filtration concept
  5. Integers object
  6. Ordered field definition
  7. Proposition concept
  8. Random experiment concept
  9. Rational numbers object
  10. Real numbers object
  11. Stopped process concept
  12. Stopping time concept
  13. Set definition
  14. Ring of integers statement
  15. Disjunction definition
  16. Implication definition
  17. Predicate definition
  18. Ordered pair definition
  19. Union definition
  20. Sample space definition
  21. Existential quantifier definition
  22. Universal quantifier definition
  23. The rationals form a field statement
  24. Cartesian product definition
  25. Subset definition
  26. Uniqueness quantifier definition
  27. The rationals form an ordered field statement
  28. Binary relation definition
  29. Universal set definition
  30. Power set definition
  31. Order on the reals definition
  32. Function definition
  33. Complement definition
  34. Interval definition
  35. Injective function definition
  36. Preimage of a set definition
  37. Sigma-algebra definition
  38. Countable set definition
  39. Event definition
  40. Borel sigma-algebra definition
  41. Probability axioms concept
  42. Probability measure definition
  43. Probability space definition
  44. Conditional probability definition
  45. Random variable definition
  46. Discrete random variable definition
  47. Stochastic process concept
  48. Random variable with a density definition
  49. Expected value definition
  50. Conditional expectation definition
  51. Martingale definition
  52. Local martingale definition
  53. Semimartingale definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Semimartingale
About this page
TagMZNFPJJMY9
URIm2://MZNFPJJMY9
Kinddefinition
WritingW0 stub
MSC60G48, 60H05
DefinesSemimartingale
UsesFiltration, Adapted process, Local martingale
Changed2 November 2026

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